Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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1answer
29 views

Transport equations with constant coefficients

Let $X$ be the vector field given by $X = b \cdot \nabla_x + \partial_t$ where $b \in \mathbb{R}^n$ is fixed. Let $f \in C^1(\mathbb{R}^{n+1})$. Assume that $u \in C^1(\mathbb{R}^n \times ...
1
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0answers
30 views

Uniformly boundedness of convolutions

Assume $X$ is an absolutely continuous random variable with pdf $f:\mathbb{R}\to[0,\infty)$. Assume further there exists $M>0$ s.t. $|f(t)|\leq M \quad\forall t\in\mathbb{R}$. Let $X_1,\dots,X_n$ ...
-2
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0answers
48 views

$ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0$ [closed]

I need help, I dont understad how it do $ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0$ please please ...
1
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1answer
30 views

If partial derivatives w.r.t. x and y are equal at each point (x,y) then which options are correct?

Let, $f$ be a function on $\mathbb R^2$ such that $\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)$ for all $(x,y)\in \mathbb R^2$. Then which is(/are) correct? ...
0
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1answer
31 views

A function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous.

I'm having a confusion over the veracity of the statement that a function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous. I've seen from a ...
1
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1answer
41 views

$f\in BV[a,b]$ has the intermediate value property , then is it true that $f$ is continuous on $[a,b]$ ?

If $f\in BV[a,b]$ has the intermediate value property , then is it true that $f$ is continuous on $[a,b]$ ? Please help . Thanks in advacne
2
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1answer
36 views

$f:[a,b]\to \mathbb R$ is continuous , has a finite number of local maxima and minima ; then how to prove that $f$ is bounded variation on $[a,b]$ ?

If $f:[a,b]\to \mathbb R$ is a continuous function having finite number of local maxima and minima ; then how to prove that $f$ is bounded variation on $[a,b]$ ?
1
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1answer
8 views

Change of Basis in Canonical Correlation Analysis

I am studying canonical correlation analysis. And I'm completely stumped for the last few days at the following manipulation. How does the following change of basis works? The equation doesn't even ...
1
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1answer
22 views

Characterizing connected sets of $\mathbb R^n$ is terms of differentiable maps for which zero derivative everywhere implies constant

Let $U $ be an open subset of $ \mathbb R^n$ ; then how to prove that $U$ is connected iff for every differentiable function $f:U \to \mathbb R$ , $\nabla f(x)=0 \implies f $ is constant on $U$ ?
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0answers
11 views

How to rescale parameters?

First of all, I am a maths newby and never got any education on rescaling parameters on whatsoever. The knowledge that I have is based on what I know from mathematical research papers and as ...
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0answers
21 views

Help with a definition involving multiple suprema/infima

I have trouble understanding a definition that comes up in a proof of the Prokhorov theorem. Let $E$ be a Polish space and $M$ a set of probability measures on the Borel $\sigma$-algebra on $E$. From ...
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0answers
7 views

Weight change of other criteria in sensitivity analysis

I want to conduct a simple senstivity analysis as described here (page 45). Let's assume I have three criteria: C1, C2 and C3. I weighted them 50%, 30%, 20%. Now I want to drive the sensitivity ...
1
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1answer
25 views

$P$ is a monic polynomial of degree $n$ , then which are correct?

Suppose that $P$ is a monic polynomial of degree $n$ in one variable with real coefficients and $K$ is a real number. Then which of the following statements are necessarily correct ? If $n$ ...
0
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1answer
28 views

If $f\in C^2(\mathbb R)$ then $M_1^2 \le 2M_0 M_2$, where $M_k = \text {sup}_x |(d/dx)^k f(x)|$ for $k=0,1,2.$

I wanna prove this problem. I tried it with Mean Value Theorem but cannot proceed to any plausible result. So could I have some hints?
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1answer
69 views

Find the minimum value of $P=\frac{1}{2-x}+\frac{1}{2-y}+\frac{1}{2-z}$

Let $x,y,z$ be positive real numbers such that $x^3+y^3+z^3=3$. Find the minimum value of $$P=\frac{1}{2-x}+\frac{1}{2-y}+\frac{1}{2-z}.$$ I think that we need to show that $\dfrac{1}{2-x} \ge ...
1
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1answer
34 views

What does bounded partial derivatives exactly mean?

This might be a naive question, but if I give myself a continuously differentiable function $f$ from $\mathbb{R}^n$ to $\mathbb{R}$ which is said to have bounded partial derivatives, does this mean ...
4
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0answers
25 views

Why is n-th Fréchet derivative symmetric?

Let $V,W$ be nonzero normed spaces over $\mathbb{K}$. Let $E$ be open in $V$ and $f:E\rightarrow W$ be a twice Fréchet-differentiable function. Then, $D^2 f: E\rightarrow \mathscr{L}_2(V^2,W)$ is ...
4
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2answers
130 views

Solving ODE rigorously

I am given the ODE $$(f''(r)+\frac{f'(r)}{r})(1+f'(r)^2)-f'(r)^2f''(r)=0$$ and want to solve it rigorously for $r>0.$ So especially, I don't want to loose any solutions. $\textbf{Derivation of ...
0
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1answer
39 views

If $u=e^x \cos y \text{ and } v=e^x \sin y$ transform the following: $w_{xx}+w_{yy}=0.$

If $u=e^x \cos y \text{ and } v=e^x \sin y$ transform the following: $w_{xx}+w_{yy}=0.$ I was hoping that someone would maybe be familiar to this $w$ function that is stated, because this is the only ...
2
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3answers
171 views

Are strongly equivalent metrics mutually complete?

Maybe I'm missing something, but I can't seem to find any references to my exact question. If two metrics, $d_1(x,y)$ and $d_2(x,y)$ are strongly equivalent, then there exists two positive constants, ...
0
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1answer
26 views

Continuous functions with domain in the Natural Numbers

Can functions with domain in the Natural Numbers be continuous? In the high school, it is teached an intuitive notion of continuous functions: functions which will always appear as an "unbroken ...
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0answers
17 views

Construction of a function with linear start and horizontal asymptote equal to 1

I need a function which starts linearly at x=0 (with parameter settable slope !) and approaches horizontal asymptote of y=1 if x goes to infinity. Also the functions convergence speed to the asymptote ...
1
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1answer
35 views

Properties of decreasing sequence of Lebesgue measurable sets.

I'm trying to prove a property of Lebesgue measure sets that says: If the $A_{k}$'s are measurable and $A_{1} \supset A_{2} \supset A_{3} \supset \ldots,$ and if $\lambda (A_{1}) < \infty, $ then ...
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2answers
34 views

Prove that If $\lim_{n \to \infty} |x_n-x_{n+p}| = 0$ for all $p \geq 1$, then $\{x_n\}$ is Cauchy sequence ??? [closed]

Let $\{x_n\}$ be a sequence in $\mathbb{R}$. If $\lim_{n \to \infty} |x_n-x_{n+p}| = 0$ for all $p \geq 1$, then $\{x_n\}$ is a Cauchy sequence.
1
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1answer
24 views

How to show that $D \det_A (H)$ exists and equals $\det( adj(A)H)$?

Consider the function $\det : M_n(\mathbb R) \to \mathbb R$ ; how to show that for any $A , H \in M_n(\mathbb R)$ , the derivative operator of determinat of $A$ evaluated at $H$ i.e. $D \det_A (H)$ ...
2
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0answers
62 views

Measuring the set-theoretical complexity of sets/spaces encountered in general analysis

In analysis, it is common to encounter subsets of $\mathbb R$ (or even $\mathbb R^n$) which appear to be "well-behaved", especially with regard to properties like being measurable, compactness, etc. ...
2
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0answers
46 views

Example of Measure of non-compactness?

I can't understand the following example of measure of non-compactness, which was given in a research article. Definition: A nonnegative function $\phi$ defined on the bounded subsets of $X$ will ...
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0answers
38 views

Can we make $S_n \to \delta_x$ for $S_n$ an exponential polynomial?

Consider $f_\lambda: \Bbb{R}_+ \to \Bbb{R}_+$,$$f_\lambda(t) = e^{-\lambda t}$$ Now consider the finite linear combinations of these functions (exponential polynomials) $$ S(t) = \sum_{i = 1}^N ...
1
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2answers
50 views

Proof that Right hand and Left hand derivatives always exist for convex functions.

Definition A function $f$ is convex on an interval if for $a,x, \text{and} \;b$ in the interval with $a\lt x\lt b$, we have $$\frac{f(x)-f(a)}{x-a}\lt \frac{f(b)-f(a)}{b-a}.$$ While reading the ...
0
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1answer
19 views

finding sup,inf, max,min of $A\cap B$ and $A\cup B$, if they exist and proving that $A\cap B$ is bounded.

As the title says i am trying to find and prove inf,sup, min, max if they exist for $A\cup B$ and $A\cap B$. And then prove that $A\cap B$ is bounded. Which will actually be easy, after i find all ...
1
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1answer
72 views

How to prove by Mathematical Induction. [duplicate]

I want to know how to prove this inequality by mathematical induction: $a_k's$ are nonnegative numbers. Prove that$$a_1a_2\cdots a_n\leq \left(\frac{a_1+a_2+\cdots+a_n}{n}\right)^n.$$ In the inductive ...
0
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2answers
28 views

finding for which $c\in \mathbb{R}$ sequence converges

so i am trying to find for which $c\in\mathbb{R}$ this sequence converges: $a_{1}=c$ and $a_{n+1}=1+\frac{a_{n}^2}{4}$ So i got the basic idea how to do this. First i found the candidate for limit: ...
0
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1answer
36 views

Finding a function such that $(n-2)/2 + f(n-1) \leq f(n)$

By bounding a certain quantity defined on real numbers by $f(n)$ I derived the following inequality arising from an inductive argument. $ (n-2)/2 + f(n-1) \leq f(n).$ A solution to the above ...
0
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1answer
39 views

$t\in (0,1)$ and $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$. Show that if strict inequality holds for even one $t$, then it holds for all $t$.

This is a part of a solution to a problem in showing that if $f$ is continuous and satisfies the condition $f([x+y]/2)\lt [f(x)+f(y)]/2$, then $f$ is convex. Let $t\in (0,1)$. We have the weak ...
1
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1answer
28 views

Verification of a Diffeomorphism

Below is an exercise to prepare for an Analysis II Exam Let $f: \mathbb{R} \to \mathbb{R}$ be a function of Class $C^1$ such that $|f'(t)| \leq k < 1$ for all $t \in \mathbb{R}$. Show that the ...
0
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1answer
38 views

Local extremes of $f(x) = (x-2)^{\frac{1}{5}}(x-7)^{\frac{1}{9}}$

The task is to find local extremes of $f: \mathbb R \to \mathbb R$, $f(x) = (x-2)^{\frac{1}{5}}(x-7)^{\frac{1}{9}}$ There is theorem that if $x_{0}$ is local extreme of $f(x)$ then $f'(x_0) = 0$ So ...
1
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1answer
41 views

Difficulties of convergence of the partial sums of the Fourier inversion formula

Define the Fourier inversion formula over $\mathbb{R}^n$ by $$ f(x)=\int_{\mathbb{R}^n}e^{2\pi ix\cdot\xi}\hat{f}d\xi $$ where $\hat{f}$ is the Fourier transform over $\mathbb{R}^n$ $$ ...
3
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2answers
125 views

Is $f(x) = x^3 \sin \frac{1}{x} $ uniformly continuous on $(0, \infty)$?

Since the derivitive of $f$ is bounded on a neighborhood of $0$, $f$ is uniformly continuous on $(0, M)$ where $M$ is any positive number. I'd like to prove that $f$ is uniformly continuous on a ...
0
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1answer
25 views

Series of positive-definite kernels

Suppose I have a positive definite, shift invariant kernel $k_1(x-y)=k_1(\delta)$. I want to know whether the sum (where $a_n\geq 0$) $$ k(\delta) = \sum_{n=1}^{\infty} a_n k_1(n\delta)\tag{*} $$ is ...
2
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1answer
57 views

Prove that the set of square matrices $A(x)=\begin{pmatrix} 2x+y & x \\ 3x & 2x+3y \\ \end{pmatrix}$ for $x,y\in [0,1]$ is a compact set.

Prove that the set of square matrices $A(x)=\begin{pmatrix} 2x+y & x \\ 3x & 2x+3y \\ \end{pmatrix}$ for $x,y\in [0,1]$ is a compact set.(Take into consideration metric $d_2...$) I was ...
3
votes
2answers
177 views

Does anyone have a proof that the intersection and union of two compact sets is compact.

I have my take on it. It is quite informal and don;t know where it would be evaluated correctly on an exam. Since the sets are compact that means for every open cover there is a finite cover. When ...
1
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1answer
41 views

Examine the uniform convergence of the series $\sum^{\infty}_{n=1}\frac{1}{\sqrt{x+n}}$ if $x \in [0, \infty]$

Examine the uniform convergence of the series $$\sum^{\infty}_{n=1}\frac{1}{\sqrt{x+n}}$$ if $x \in [0, \infty)$ Which series should I choose in Weierstrass M-test to show that is divergent? ...
0
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1answer
32 views

What are the maps of these closed sets in $\mathbb R^3 \mapsto \mathbb R$

What is the map of an elipsoid(closed) if $H(x,y,z)=x+y+z$ This is kind of an tricky question, because I am not sure precisely what the answer is. I think when it says closed elipsoid it can be ...
1
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0answers
31 views

Proof of Gruss inequality

I've been reading articles that use the Gruss inequality for some time now, but I can't seem to find a proof of it anywhere. The only source I could find that actually has the proof is the original ...
0
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0answers
15 views

For what does the formula $(\prod_{t=1}^d[\begin{array}{c}-\frac 12&1&-\frac 12\end{array}]_{l_t,i_t})f$ stand for?

Let $f:\mathbb R\to\mathbb R$ and $$a_{l,i}:=f(x_{l,i})-\frac{f(x_{l-1,(i-1)/2}+f(x_{l-1,(i+1)/2})}2$$ for some $x_{l,i}$. I've read, that we can write $a_{l,i}$ in the following "operator form": ...
2
votes
1answer
66 views

What does the term “regularity” mean?

When I was an undergraduate, I took a course on regularity theory for nonlinear elliptic systems. This included topics such as the direct method of calculus of variations, mollifiers, integration over ...
1
vote
1answer
35 views

If a compact subset is contained in an open subset in $\mathbb{R}^n$, is a small cylinder of this compact subset also contained in the open set?

Let $O\subseteq \mathbb{R}^n$ be an open set, $K\subseteq \mathbb{R}^{n-1}$ a compact set and $a\in \mathbb{R}$, such that $$\{a\}\times K\subseteq O$$ holds. Does there exist an $\epsilon>0$, ...
-1
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0answers
33 views

Prove the norm axioms

Prove the norm axioms for example 7 . Thank you. :)
0
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1answer
20 views

Prove that $\left|\frac{x^p-1}{p}\right|\leq x+|\ln(x)|$ for all $x\in(0,\infty)$ and for all $p\in(0,1)$

So far I have shown that $$\displaystyle \lim\limits_{p\to 0^+}\frac{x^p-1}{p}=\lim\limits_{p\to 0^+}\frac{e^{p\ln(x)}-1}{p}=\ln(x)\lim\limits_{p\to 0^+}\frac{e^{p\ln(x)}-1}{p\ln(x)} =$$ (L'Hopital) ...
2
votes
2answers
63 views

$\mu(A \cap I) \le a \mu(I)$ implies $\mu(A) = 0$?

Let $\mu$ be lebesgue measure on $\mathbb{R}$, $0<a<1$. If $\mu(A \cap I) \le a \mu(I)$ holds for any interval $I$, can I say $\mu(A)=0$? I tried to construct a counterexample by considering ...